The sheer sizes of modern datasets are forcing data-structure designers to consider seriously both parallel construction and compactness. To achieve those goals we need to design a parallel algorithm with good scalability and with low memory consumption. An algorithm with good scalability improves its performance when the number of available cores increases, and an algorithm with low memory consumption uses memory proportional to the space used by the dataset in uncompact form. In this work, we discuss the engineering of a parallel algorithm with linear work and logarithmic span for the construction of the compact representation of planar embeddings. We also provide an experimental study of our implementation and prove experimentally that it has good scalability and low memory consumption. Additionally, we describe and test experimentally queries supported by the compact representation.